A051612 -id:A051612 - cp's OEIS frontend

A110085 Numbers k such that sigma(k) - phi(k) < tau(k)^omega(k), where sigma = A000203, phi = A000010, tau = A000005 and omega = A001221.

1, 6, 10, 12, 18, 20, 24, 30, 36, 42, 60, 66, 70, 78, 84, 90, 102, 105, 110, 114, 120, 126, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 180, 182, 186, 190, 195, 198, 204, 210, 220, 222, 228, 230, 231, 234, 238, 240, 246, 252, 255, 258, 260, 264, 266

Offset: 1

Reinhard Zumkeller, Jul 11 2005

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Intersection of A018252 and A110086

Cf. A000005, A000010, A000203, A001221, A051612, A110087, A110088.

a110085 n = a110085_list !! (n-1)
a110085_list = filter (\x -> a051612 x < a110088 x) [1..]
-- Reinhard Zumkeller, Aug 05 2014

q[k_] := DivisorSigma[1, k] - EulerPhi[k] < DivisorSigma[0, k]^PrimeNu[k]; Select[Range[300], q] (* Amiram Eldar, Sep 15 2024 *)

is(n)=sigma(n)-eulerphi(n)Charles R Greathouse IV, Feb 14 2013

A051612(a(n)) < A110088(a(n)).

A110087 Numbers k such that sigma(k) - phi(k) > tau(k)^omega(k), where sigma = A000203, phi = A000010, tau = A000005 and omega = A001221.

Original entry on oeis.org

4, 8, 9, 14, 16, 21, 22, 25, 26, 27, 28, 32, 33, 34, 35, 38, 39, 40, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 64, 65, 68, 69, 72, 74, 75, 76, 77, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104, 106, 108, 111, 112, 115, 116, 117, 118

Offset: 1

Reinhard Zumkeller, Jul 11 2005

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Complement of A110086.

Cf. A000005, A000010, A000203, A001221, A051612, A110088.

a110087 n = a110087_list !! (n-1)
a110087_list = filter (\x -> a051612 x > a110088 x) [1..]
-- Reinhard Zumkeller, Aug 05 2014

q[k_] := DivisorSigma[1, k] - EulerPhi[k] > DivisorSigma[0, k]^PrimeNu[k]; Select[Range[120], q] (* Amiram Eldar, Sep 15 2024 *)

is(n)=sigma(n)-eulerphi(n)>numdiv(n)^omega(n) \\ Charles R Greathouse IV, Feb 19 2013

A051612(a(n)) > A110088(a(n)).

A240960 Numbers m such that sigma(m) - phi(m) = tau(m)^omega(m), where sigma=A000203, phi=A000010, tau=A000005 and omega=A001221.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Reinhard Zumkeller, Aug 05 2014

nonn

a(n) = A182140(n) for n <= 35.

All primes p are in the sequence since (p+1) - (p-1) = 2^1. The first composites are 15, 119748396, 139254850, 187768485, 1420027536, 3991789984. A182140 seems unrelated. - Jens Kruse Andersen, Aug 05 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000203, A001221, A051612, A110088, A182140, A110086.

a240960 n = a240960_list !! (n-1)
a240960_list = filter (\x -> a051612 x == a110088 x) [1..]

with(numtheory):
filter:= n -> sigma(n) - phi(n) = tau(n)^nops(factorset(n)):
select(filter, [$1..1000]); # Robert Israel, Aug 05 2014

Select[Range[300], DivisorSigma[1, #] - EulerPhi[#] == DivisorSigma[0, #]^PrimeNu[#]&] (* Jean-François Alcover, Mar 08 2019 *)

is(n)=my(f=factor(n)); sigma(f)-eulerphi(f)==numdiv(f)^omega(f) \\ Charles R Greathouse IV, Nov 26 2014

from sympy import totient,divisors,divisor_count,primefactors
filter(lambda x:sum(divisors(x))-totient(x)==divisor_count(x)**len(primefactors(x)), range(1,10**5)) # Chai Wah Wu, Aug 05 2014

A292786 a(n) = psi(n) - phi(n).

Original entry on oeis.org

0, 2, 2, 4, 2, 10, 2, 8, 6, 14, 2, 20, 2, 18, 16, 16, 2, 30, 2, 28, 20, 26, 2, 40, 10, 30, 18, 36, 2, 64, 2, 32, 28, 38, 24, 60, 2, 42, 32, 56, 2, 84, 2, 52, 48, 50, 2, 80, 14, 70, 40, 60, 2, 90, 32, 72, 44, 62, 2, 128, 2, 66, 60, 64, 36, 124, 2, 76, 52, 120, 2, 120, 2, 78, 80, 84, 36, 144

Offset: 1

Altug Alkan, Sep 23 2017

Even numbers that are not the terms of this sequence are 12, 102, 114, 130, ...

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A001615, A051612, A069359, A070915, A088245, A291784.

psi[n_] := If[n < 1, 0, n Sum[ MoebiusMu[d]^2/d, {d, Divisors@ n}]]; Array[psi@# - EulerPhi@# &, 87] (* Robert G. Wilson v, Sep 23 2017 *)

a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
a(n) = a001615(n) - eulerphi(n); \\ after Charles R Greathouse IV at A001615

a(n) = A001615(n) - A000010(n).
a(n) = 2 iff n is prime.
a(n) = 2*A069359(n) iff n is in A070915.
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c =  9/(2*Pi^2) = 0.455945... (A088245). - Amiram Eldar, Dec 05 2023

A071390 Least number m such that sigma(m) - phi(m) = n, or 0 if no such m exists.

Original entry on oeis.org

0, 2, 0, 0, 4, 0, 9, 0, 0, 6, 8, 0, 0, 10, 49, 15, 0, 14, 0, 21, 0, 27, 16, 12, 0, 22, 169, 33, 0, 26, 0, 39, 18, 20, 289, 65, 0, 34, 361, 51, 0, 38, 0, 28, 0, 0, 32, 95, 0, 46, 0, 24, 0, 45, 0, 115, 0, 0, 841, 161, 0, 58, 961, 30, 0, 62, 81, 63, 0, 0, 0, 155, 50, 40, 1369, 217, 0, 74

Offset: 1

Labos Elemer, May 23 2002

nonn

For n <> 2, a(n) < n^2/4. - Robert Israel, Apr 02 2020

			n=255: a(255) = 16129 = 127^2, sigma(16129) = 16257, phi(16129) = 16002, 16257 - 16002 = 255 = n. Squares of primes are often solutions (4, 9, 49, 169, 289, 361, etc.).

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A000010, A000203, A065387, A051612, A071391.

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for m from 2 to N^2/4 do
  v:= numtheory:-sigma(m)-numtheory:-phi(m);
  if v <= N and V[v]=0 then V[v]:= m fi
od:
convert(V,list); # Robert Israel, Apr 02 2020

f[x_] := DivisorSigma[1, x]-EulerPhi[x] t=Table[0, {100}]; Do[c=f[n]; If[c<101&&t[[c]]==0, t[[c]]=n], {n, 1, 1000}]; t

a(n) = Min{x; A000203(x)-A000010(x)=n} or a(n)=0 if no solution exists.

A270836 Numbers k such that sigma(k-1) - phi(k-1) = (3*k-5)/2.

Original entry on oeis.org

3, 5, 9, 11, 17, 33, 65, 129, 231, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 119831, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649, 4294967297

Offset: 1

Jaroslav Krizek, Mar 23 2016

nonn

Numbers k such that A051612(k-1) = (3*k-5)/2.
Numbers of the form 2^k + 1 for k >= 1 from A000051 are terms.
Prime terms are in A270778.

			17 is a term because sigma(16) - phi(16) = 31-8 = 23 = (3*17-5)/2.

Giovanni Resta, Table of n, a(n) for n = 1..46 (terms < 10^13)

Cf. A000010, A000051, A000203, A051612, A270778, A270837.

[n: n in[2..10^7] | 2*(SumOfDivisors(n-1) - EulerPhi(n-1)) eq 3*n-5];

Select[Range[10^6], 2 (DivisorSigma[1, # - 1] - EulerPhi[# - 1]) == 3 # - 5 &] (* Michael De Vlieger, Mar 24 2016 *)

lista(nn) = {for(n=2, nn, if(sigma(n-1) - eulerphi(n-1) == (3*n-5)/2, print1(n, ", "))); } \\ Altug Alkan, Mar 23 2016

a(30)-a(32) from Michel Marcus, Apr 05 2016

a(33)-a(35) from Giovanni Resta, Apr 11 2016

A036446 Not of form sigma(k) - phi(k).

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 12, 13, 17, 19, 21, 25, 29, 31, 37, 41, 43, 45, 46, 49, 51, 53, 55, 57, 58, 61, 65, 69, 70, 71, 77, 81, 82, 85, 89, 91, 93, 97, 99, 101, 103, 105, 109, 111, 113, 114, 115, 117, 118, 121, 125, 127, 130, 131, 133, 134, 137, 139, 141, 142, 145, 149, 151

Offset: 1

David W. Wilson

nonn

Robert Israel, Table of n, a(n) for n = 1..5510 (terms <= 10000)

Complement of A112730.

Cf. A000010, A000203, A051612.

A061367 Composite n such that sigma(n)-phi(n) divides sigma(n)+phi(n).

Original entry on oeis.org

15, 35, 95, 119, 143, 209, 287, 319, 323, 357, 377, 527, 559, 779, 899, 923, 989, 1007, 1045, 1189, 1199, 1343, 1349, 1763, 1919, 2159, 2261, 2507, 2639, 2759, 2911, 3239, 3339, 3553, 3599, 3827, 4031, 4147, 4607, 5049, 5183, 5207, 5249, 5459, 5543, 6439

Offset: 1

Joseph L. Pe, Feb 13 2002

nonn

Primes trivially satisfy the defining condition.

			sigma(15)-phi(15) = 24-8 = 16 divides sigma(15)-phi(15)=24+8 = 32, so 15 is a term of the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A051612, A065387.

f[n_] := Module[{a = DivisorSigma[1, n], b = EulerPhi[n]}, Mod[(a + b), (a - b)] == 0]; Select[Range[2, 10^4], (f[ # ] && ! PrimeQ[ # ]) &]
cnQ[n_]:=With[{s=DivisorSigma[1,n],p=EulerPhi[n]},Mod[s+p,s-p]==0]; Select[Range[6500],CompositeQ[#]&&cnQ[#]&] (* Harvey P. Dale, Jun 14 2025 *)

It seems that a(n) is asymptotic to c*n^2, 22*n^2. - Benoit Cloitre, Sep 17 2002

A071391 Least number m such that sigma(m) + phi(m) = n or 0 if no such number exists.

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 0, 4, 5, 0, 0, 0, 6, 0, 0, 0, 0, 8, 0, 0, 10, 0, 0, 0, 13, 0, 0, 0, 14, 0, 12, 0, 17, 0, 0, 0, 19, 16, 0, 0, 0, 0, 21, 18, 22, 0, 0, 0, 20, 25, 0, 0, 26, 0, 0, 0, 27, 0, 0, 0, 31, 0, 0, 0, 0, 0, 24, 0, 34, 0, 35, 0, 37, 0, 0, 0, 38, 32, 30, 0, 41, 0, 0, 0, 43, 0, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, May 23 2002

			n=256: a(256) = 110, sigma(110) + phi(110) = 216 + 40 = 256 = n and no positive integer k < 110 has sigma(k) + phi(k) = 256.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A065387, A051612, A071390.

f[x_] := DivisorSigma[1, x]+EulerPhi[x] t=Table[0, {100}]; Do[c=f[n]; If[c<101&&t[[c]]==0, t[[c]]=n], {n, 1, 1000000}]; t

a(n) = for(m=1, n, if(sigma(m)+eulerphi(m)==n, return(m))); 0; \\ Jinyuan Wang, Jul 29 2020

first(n) = { my(v = vector(n)); for(i = 1, n, c = sigma(i) + eulerphi(i); if(c <= n, if(v[c] == 0, v[c] = i ) ) ); v } \\ David A. Corneth, Jul 30 2020

a(n) = Min{x; A000203(x)+A000010(x)=n} or a(n) = 0 if no solution exists.

A077088 a(n) = phi(sigma(n) - phi(n)), where phi is Euler's totient function and sigma is the sum of divisors function, with a(1) = 0.

Original entry on oeis.org

0, 1, 1, 4, 1, 4, 1, 10, 6, 6, 1, 8, 1, 6, 8, 22, 1, 20, 1, 16, 8, 12, 1, 24, 10, 8, 10, 20, 1, 32, 1, 46, 12, 18, 8, 78, 1, 12, 16, 36, 1, 24, 1, 32, 18, 20, 1, 36, 8, 72, 16, 36, 1, 32, 16, 32, 20, 30, 1, 72, 1, 20, 32, 72, 12, 60, 1, 46, 24, 32, 1, 108, 1, 24, 24, 48, 12, 48, 1, 60

Offset: 1

Labos Elemer, Nov 04 2002

a(p) = 1 for p prime. Otherwise a(n) is even.

			a(10) = 6 because sigma(10) = 18 and phi(10) = 4, and so phi(18 - 4) = phi(14) = 6.
a(11) = 1 because sigma(11) = 12 and phi(11) = 10, so phi(12 - 10) = phi(2) = 1.
a(12) = 8 because sigma(12) = 28 and phi(12) = 4, so phi(28 - 4) = phi(24) = 8.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A000203, A051612, A065387. See iterations in A077090-A077100.

List([1..100],n->Phi(Sigma(n)-Phi(n))); # Muniru A Asiru, Mar 04 2018

with(numtheory); A077088:=n->phi(sigma(n)-phi(n)); seq(A077088(n), n=1..100); # Wesley Ivan Hurt, Dec 02 2013

Table[EulerPhi[DivisorSigma[1, n] - EulerPhi[n]], {n, 100}] (* Alonso del Arte, Nov 29 2013 *)

A077088(n) = if(1==n,0,eulerphi(sigma(n) - eulerphi(n))); \\ Antti Karttunen, Mar 04 2018

a(1) = 0; and for n > 1, a(n) = A000010(A051612(n)).

Value of a(1) clarified by Antti Karttunen, Mar 04 2018

cp's OEIS Frontend

A110085 Numbers k such that sigma(k) - phi(k) < tau(k)^omega(k), where sigma = A000203, phi = A000010, tau = A000005 and omega = A001221.

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A110087 Numbers k such that sigma(k) - phi(k) > tau(k)^omega(k), where sigma = A000203, phi = A000010, tau = A000005 and omega = A001221.

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A240960 Numbers m such that sigma(m) - phi(m) = tau(m)^omega(m), where sigma=A000203, phi=A000010, tau=A000005 and omega=A001221.

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A292786 a(n) = psi(n) - phi(n).

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A071390 Least number m such that sigma(m) - phi(m) = n, or 0 if no such m exists.

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A270836 Numbers k such that sigma(k-1) - phi(k-1) = (3*k-5)/2.

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A036446 Not of form sigma(k) - phi(k).

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A061367 Composite n such that sigma(n)-phi(n) divides sigma(n)+phi(n).

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